Multivariable Calculus/Calculus & Geometry textbook suggestions

  1. Hello all,

    I am currently in University and I am retaking a course that I dropped last year due to the fact that there was no textbook and the time I spent in the course was a frantic scramble for reliable information and attempting to understand what the professor was teaching. I have looked before and the only textbook that I found that was remotely close to the material that is covered in the course is Multivariable Calculus by Edwards and Penney. I am asking the community for any suggestions for textbooks that cover most, if not all of the topics listed below:

    - Differentiation in Several Variables
    Partial derivatives, directional derivatives, del operator, Mean-value theorem for a function of several variables, differentiation through an integral, Leibniz's Rule 2nd order derivatives and Clairaut's theorem, Hessian Matrix. Functions from several variables to several variables, Jacobian Matrix

    - Inverse - and Implicit functions in several variables
    Inverse function theorem
    Implicit function theorem

    - Quadratic forms
    Matrix Representation
    Change of variable & diagonalization by congruence
    Positive & Negative definiteness, Semi-definieness, determinantal criteria
    Sylvester's Theorem

    - Extrema
    Local extrema-critical points, local max, local min, saddle points, determinantal tests on the Hessian Matrix
    Global extrema-existence for continuous functwions on closed, bounded sets, finiding constrained extrema by the method of Lagrange multipliers

    - Curves and Surfaces
    Space curves defined parametrically-tangent, normal, binormal, arc length, curvature, torsion
    Surfaces-implicit and parametric definitions-tangent plane, normal line
    Space curves defined as intersecting surgaces, related quadratic & cubic forms

    - Integration in several Variables
    Multiple and iterated integrals
    Change of order of variables
    Change of variable formula in general
    Polar, cylindrical & spherical coordinates
    Line integrals, consecutive and non-consecutive vector fields, curl operator
    Surface integrals, projection onto a plane, parametric surgace integrals
    Stokes' theorem, boundary curve, orientation
    Gauss' theorem, boundary surgace, div operator

    Any suggestions will be greatly appreciated and sorry for the long list, I didn't want to leave anything out.
  2. jcsd
  3. jbunniii

    jbunniii 3,377
    Science Advisor
    Homework Helper
    Gold Member

  4. I fully second Apostol and Courant.
  5. Apostol and Courant are excellent but they are mathematically rigorous and I don't know if that is what you want. If not, then there is the book by Schey Div, Grad, Curl and All That for intuition; Marsden and Tromba Vector Calculus is an intermediate sort of book; and the old favourite is Stewart Multivariable Calculus or the relevant chapters in his Calculus. Stewart gives the simplest presentation of calculus.

    I suggest going to the library and looking through all these books and deciding for yourself which one best suites your needs.
  6. any multivariable calculus text will fit your description..
  7. I agree that Courant and Apostol are NOT light reads. Stewart has an excellent MV book that is good for all audiences. A lesser known book is Howard Anton's Calculus, A new horizon, which has a lot of good applications of calc added in. Of course the Khan Academy has very easy-to-understand lectures, but not in all of the above mentioned topics.
  8. eumyang

    eumyang 1,347
    Homework Helper

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